I heard a talk recently on number theoretic representation theory, in which the speaker suggested that we focus on the case $G=$GL$_2$, with $\rho$, the representations, being thought of as sym$^n$ for simplicity.
This is all kind of new to me, so I'm rather confused. In this kind of generality and without an ability to furnish more context than I've already given, is it clear what GL$_2$ is? I guess one fixes some kind of a field of coefficients, but I'm not sure.
And more mysterious to me is what exactly the above representation is. Is this standard notation? From what I gather it has something to do with the symmetric power of a representation, but then it seems to me one would have to have an underlying representation first before one can take symmetric powers of it. Or am I on the wrong path completely and is sym$^n$ some kind of a specific representation of GL$_2$?
I hope this is an appropriate question.